Identify the name associated with the total strain energy theory of failure at the elastic limit (choose the classical attribution used in mechanics of materials).

Introduction / Context:
Failure theories extend uniaxial yield data to multiaxial stress states. Each classical theory corresponds to a distinct failure criterion, and many are named for the researchers who developed them. Recognizing the correct attributions helps avoid mixing up criteria during design checks.

Given Data / Assumptions:

• Total strain energy theory is referenced.
• We are at the elastic limit under multiaxial stress.
• Names provided correspond to well-known criteria.

Concept / Approach:
Total strain energy theory (sometimes attributed to Beltrami–Haigh) posits that failure occurs when the total elastic strain energy per unit volume reaches the same value as at yielding in simple tension. It differs from the distortional (shear) energy theory of Von Mises and the maximum shear stress (Tresca/Guest) theory.

Step-by-Step Solution:
1) Map theories to names: Rankine → maximum principal stress; St. Venant → maximum principal strain; Tresca/Guest → maximum shear stress; Von Mises → distortion (shear) energy; Haigh → total strain energy.2) Select Haigh's theory as the correct attribution for total strain energy.

Verification / Alternative check:
Mechanics references list Beltrami–Haigh energy criteria under total energy, separate from Von Mises (J2) distortional energy.

Why Other Options Are Wrong:

• Guest/Tresca: Maximum shear stress, not energy-based.
• St. Venant: Maximum principal strain criterion.
• Rankine: Maximum normal stress criterion.
• Von Mises: Distortion (shear) energy, not total energy.

Common Pitfalls:

• Confusing total energy (Haigh) with distortional energy (Von Mises).
• Assuming all energy-based theories are identical—they are not.

Final Answer:
Haig's theory.